Tangent Plane and Partial Derivatives: A Closer Look

Review tangent approximation In this section, we are going to see how to use the partial derivatives. When we have a function of two variables, x and y, then we have, actually, twodifferent derivatives: partial f, partial x, called f sub x. This is the derivative of f withrespect to x while keeping y constant. And we have the partial derivative f, alsocalled f prime, where we vary y and keep x as a constant. There is a formula for approximating what happens to both x and y when you varyeither one. The function f(x) tells us what happens when x is changed by a small amount, deltax. The function f(y) tells us how f changes when y is changed by a small amount,delta y. When both functions are combined with each other, the effects of changingboth variables simultaneously will add up with each other because you can imaginethat first x will be changed and then y will be changed. And then you will change your mind, or the other way around, it doesn't really matter.So if we change x by a certain amount, delta x, and if we change y by the amountdelta y, and let's say that we have z equals f of x, y, then that changes by an amountwhich is approximately f sub x times delta x plus f sub y times delta y. So, the intuition here is that the two effects add up. If x changes by a small amountand then y changes, then first changing x will modify f. How much does it modify f?The answer is that it depends on how quickly f changes with respect to x, which wecall f sub x, and if we change y then the rate of change of "f" when we change y is itsderivative with respect to y, which we call f sub y.So, we get this change in the value of f.

And of course, that's only an approximation formula. Actually, there would be higherorder terms involving second and third derivatives, and so on. One way to justify the formula is to consider its effect on the tangent planeapproximation. The formula allows us to calculate the tangent plane through a graphof the function f. To find the partial derivative of f with respect to x, we must examine the graph of f.We can see that if we slice the graph of f by a plane that is parallel to the xz-plane,we will obtain a function g whose derivative is fx. If x changes, z changes. The slopeof that relationship is the derivative with respect to x. If y changes, z changes. Theslope of that relationship is the derivative with respect to y. Therefore in each case,we have a line. Those lines are tangent to the surface. So now if we have two lines

tangent to the surface, well then together they determine for us the tangent plane tothe surface. So we know that the partial derivatives of f with respect to x and y are slopes, andwe can write down the equations of these lines in terms of xyz coordinates. Forexample, if a is equal to the coefficient of partial f with respect to x for a given point,then we have a line given by the following conditions. Suppose that the line z=z0+a(x−x0) is drawn in green to represent the line on aplane that intersects the parallel slice of the x-z plane. Then z will vary with x at arate that depends on a if y remains constant at y0. The line tangent to the graph of f at (x0, y0) has slope b. If we call this tangent lineT(x,y), and if we call the other line S(x,y), then together they determine the planecontaining f(x,y). The equation of a plane is given by the formula z equals z0 plus a times x minus x0plus b times y minus y0. If you look at what happens when you hold y constant andvary x, you can recover the first line.

If the value of x is held constant and the value of y varies, the graph of f willapproach its tangent plane. The approximation formula can be used to estimate howthe value of f changes if we change x and y at the same time.